Question

Перша куля масою 10 кг рухається зі швидкістю 4 м/с, а друга куля масою 4 кг – зі швидкістю 12 м/с. Визначити швидкість куль після їх непружного зіткнення, якщо вони рухалися до удару: 1) назустріч; 2) друга куля наздоганяла першу.

Answer 1

Ответ:

Объяснение:

1) Удар назустріч:

Перед ударом перша куля має масу 10 кг і швидкість 4 м/с, а друга куля має масу 4 кг і швидкість -12 м/с (напрямок швидкості змінюється внаслідок удару назустріч).

Застосуємо закон збереження імпульсу:

(маса₁ * швидкість₁) + (маса₂ * швидкість₂) = (маса₁ + маса₂) * швидкість

(10 кг * 4 м/с) + (4 кг * (-12 м/с)) = (10 кг + 4 кг) * швидкість

40 кг*м/с - 48 кг*м/с = 14 кг * швидкість

-8 кг*м/с = 14 кг * швидкість

швидкість = (-8 кг*м/с) / 14 кг

швидкість ≈ -0.57 м/с

Таким чином, після непружного зіткнення швидкість куль становитиме приблизно -0.57 м/с.

2) Друга куля наздоганяла першу:

Перед ударом перша куля має масу 10 кг і швидкість 4 м/с, а друга куля має масу 4 кг і швидкість 12 м/с (обидві кулі рухаються в одному напрямку).

Знову застосуємо закон збереження імпульсу:

(маса₁ * швидкість₁) + (маса₂ * швидкість₂) = (маса₁ + маса₂) * швидкість

(10 кг * 4 м/с) + (4 кг * 12 м/с) = (10 кг + 4 кг) * швидкість

40 кг*м/с + 48 кг*м/с = 14 кг * швидкість

88 кг*м/с = 14 кг * швидкість

швидкість = 88 кг*м/с / 14 кг

швидкість ≈ 6.29 м/с

Таким чином, після непружного зіткнення швидкість куль становитиме приблизно 6.29 м/с.

Answer 2

Problem Analysis

We are given two balls with different masses and velocities before a perfectly inelastic collision. We need to determine the final velocity of the balls after the collision in two scenarios: 1) when the balls are moving towards each other, and 2) when the second ball is chasing the first ball.

Solution

To solve this problem, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.

The momentum of an object is given by the product of its mass and velocity: momentum = mass × velocity.

Let's denote the mass of the first ball as m1, its initial velocity as v1, and its final velocity after the collision as vf1. Similarly, let's denote the mass of the second ball as m2, its initial velocity as v2, and its final velocity after the collision as vf2.

Scenario 1: Balls moving towards each other

In this scenario, the balls are moving towards each other before the collision. The total momentum before the collision is the sum of the momenta of the two balls:

Total momentum before collision = m1 × v1 + m2 × v2.

After the collision, the balls stick together and move with a common final velocity vf. Therefore, the total momentum after the collision is the product of the total mass (m1 + m2) and the final velocity vf:

Total momentum after collision = (m1 + m2) × vf.

According to the principle of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision:

m1 × v1 + m2 × v2 = (m1 + m2) × vf.

We can rearrange this equation to solve for vf:

vf = (m1 × v1 + m2 × v2) / (m1 + m2).

Now we can substitute the given values into this equation to find the final velocity after the collision.

Scenario 2: Second ball chasing the first ball

In this scenario, the second ball is chasing the first ball before the collision. The total momentum before the collision is the difference between the momenta of the two balls:

Total momentum before collision = m1 × v1 - m2 × v2.

After the collision, the balls stick together and move with a common final velocity vf. Therefore, the total momentum after the collision is the product of the total mass (m1 + m2) and the final velocity vf:

Total momentum after collision = (m1 + m2) × vf.

According to the principle of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision:

m1 × v1 - m2 × v2 = (m1 + m2) × vf.

We can rearrange this equation to solve for vf:

vf = (m1 × v1 - m2 × v2) / (m1 + m2).

Now we can substitute the given values into this equation to find the final velocity after the collision.

Calculation

Let's substitute the given values into the equations to find the final velocities after the collision in both scenarios:

1) Balls moving towards each other: - m1 = 10 kg, v1 = 4 m/s - m2 = 4 kg, v2 = -12 m/s (negative sign indicates opposite direction)

vf = (m1 × v1 + m2 × v2) / (m1 + m2)

Substituting the values: vf = (10 kg × 4 m/s + 4 kg × -12 m/s) / (10 kg + 4 kg)

2) Second ball chasing the first ball: - m1 = 10 kg, v1 = 4 m/s - m2 = 4 kg, v2 = 12 m/s

vf = (m1 × v1 - m2 × v2) / (m1 + m2)

Substituting the values: vf = (10 kg × 4 m/s - 4 kg × 12 m/s) / (10 kg + 4 kg)

Now we can calculate the final velocities after the collision in both scenarios.

Answer 3

Problem Analysis

We are given two balls with different masses and velocities before a completely inelastic collision. We need to determine the final velocity of the balls after the collision in two scenarios: when they are moving towards each other and when the second ball is chasing the first ball.

Solution

To solve this problem, we can apply the principle of conservation of momentum, which states that the total momentum of an isolated system remains constant before and after a collision.

The momentum of an object is given by the product of its mass and velocity: momentum = mass * velocity.

Let's denote the mass of the first ball as m1, the mass of the second ball as m2, the initial velocity of the first ball as v1, the initial velocity of the second ball as v2, and the final velocity of the balls after the collision as vf.

Scenario 1: Balls moving towards each other

In this scenario, the first ball and the second ball are moving towards each other before the collision.

According to the conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision.

The total momentum before the collision is given by the sum of the momenta of the two balls: m1 * v1 + m2 * v2

The total momentum after the collision is given by the sum of the momenta of the two balls after the collision: m1 * vf + m2 * vf

Setting the two expressions equal to each other, we can solve for vf: m1 * v1 + m2 * v2 = (m1 + m2) * vf

Now we can substitute the given values into the equation and solve for vf.

Given: - m1 = 10 kg - v1 = 4 m/s - m2 = 4 kg - v2 = -12 m/s (since the second ball is moving in the opposite direction)

Substituting the values into the equation: 10 kg * 4 m/s + 4 kg * (-12 m/s) = (10 kg + 4 kg) * vf

Simplifying the equation: 40 kg * m/s - 48 kg * m/s = 14 kg * vf

Combining like terms: -8 kg * m/s = 14 kg * vf

Solving for vf: vf = (-8 kg * m/s) / 14 kg

Calculating the value: vf ≈ -5.71 m/s

Therefore, the final velocity of the balls after the collision, when they are moving towards each other, is approximately -5.71 m/s.

Scenario 2: Second ball chasing the first ball

In this scenario, the second ball is chasing the first ball before the collision.

Again, according to the conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision.

The total momentum before the collision is given by the sum of the momenta of the two balls: m1 * v1 + m2 * v2

The total momentum after the collision is given by the sum of the momenta of the two balls after the collision: m1 * vf + m2 * vf

Setting the two expressions equal to each other, we can solve for vf: m1 * v1 + m2 * v2 = (m1 + m2) * vf

Now we can substitute the given values into the equation and solve for vf.

Given: - m1 = 10 kg - v1 = 4 m/s - m2 = 4 kg - v2 = 12 m/s (since the second ball is moving in the same direction)

Substituting the values into the equation: 10 kg * 4 m/s + 4 kg * 12 m/s = (10 kg + 4 kg) * vf

Simplifying the equation: 40 kg * m/s + 48 kg * m/s = 14 kg * vf

Combining like terms: 88 kg * m/s = 14 kg * vf

Solving for vf: vf = (88 kg * m/s) / 14 kg

Calculating the value: vf ≈ 6.29 m/s

Therefore, the final velocity of the balls after the collision, when the second ball is chasing the first ball, is approximately 6.29 m/s.

Conclusion

In summary, the final velocities of the balls after the completely inelastic collision are approximately: 1) When the balls are moving towards each other: -5.71 m/s 2) When the second ball is chasing the first ball: 6.29 m/s.

Please let me know if I can help you with anything else.